Let \(\mathcal {H}\) be a complex Hilbert space and let A positive operator on {H}\). We obtain new bounds for the A-numerical radius of operators in semi-Hilbertian {B}_A(\mathcal {H})\) that generalize improve existing ones. Further, we estimate an upper bound \(\mathbb {A}\)-operator seminorm \(2\times 2\) matrices, where {A}=\text{ diag }(A,A)\). The obtained here generalizes earlier relate...

In this work, a refinement of the Cauchy–Schwarz inequality in inner product space is proved. A more general Kato’s or so called mixed Schwarz established. Refinements some famous numerical radius inequalities are also pointed out. As shown these refinements generalize and refine recent old results obtained literature. Among others, it proved that if $$T\in \mathscr {B}\left( {H}\right) $$ , th...

New inequalities for the numerical radius of bounded linear operators defined on a complex Hilbert space $${\mathcal {H}}$$ are given. In particular, it is established that if T operator then $$\begin{aligned} w^2(T)\le \min _{0\le \alpha \le 1} \left\| T^*T +(1-\alpha )TT^* \right\| , \end{aligned}$$ where w(T) T. The obtained here non-trivial improvement well-known inequalities. As an applica...